Nuprl Lemma : dense-in-reals-iff
∀X:ℝ ⟶ ℙ. (dense-in-interval((-∞, ∞);X) ⇐⇒ ∀x:ℝ. ∀n:ℕ+. ∃y:ℝ. ((X y) ∧ (|x - y| < (r1/r(n)))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
dense-in-interval: dense-in-interval(I;X),
true: True,
rdiv: (x/y),
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rless: x < y,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
nat_plus_wf,
real_wf,
dense-in-interval_wf,
riiint_wf,
subtype_rel_dep_function,
i-member_wf,
member_riiint_lemma,
subtype_rel_self,
set_wf,
all_wf,
exists_wf,
rless_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
true_wf,
radd_wf,
radd-preserves-rless,
rless_functionality,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
rabs-difference-bound-iff,
rless-implies-rless,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
ravg-dist,
small-reciprocal-real,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
rleq_weakening_rless,
rleq_weakening,
rmul_preserves_rless,
rminus_wf,
rmul-zero-both,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
nequal_wf,
itermMinus_wf,
radd-zero,
radd-rminus-assoc,
req_weakening,
rmul_functionality,
rabs-of-nonneg,
req_transitivity,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
real_term_value_minus_lemma,
ravg_wf,
ravg-between,
req_inversion,
rless_transitivity1,
radd-preserves-rleq,
rleq_functionality,
rless_functionality_wrt_implies,
radd_functionality_wrt_rless1,
rabs-difference-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
sqequalRule,
lambdaEquality,
universeEquality,
setEquality,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
because_Cache,
productEquality,
functionExtensionality,
natural_numberEquality,
inrFormation,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
functionEquality,
dependent_set_memberEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
minusEquality,
addLevel
Latex:
\mforall{}X:\mBbbR{} {}\mrightarrow{} \mBbbP{}. (dense-in-interval((-\minfty{}, \minfty{});X) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}y:\mBbbR{}. ((X y) \mwedge{} (|x - y| < (r1/r(n)))))
Date html generated:
2017_10_03-AM-10_10_09
Last ObjectModification:
2017_09_13-PM-00_16_06
Theory : reals
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